Definition

1. Noun (Mathematics):

   • Isomorphism: A structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. It indicates that the two structures are essentially the same in terms of their form or operations, even if their elements are different.

2. Adjective (Mathematics):

   • Isomorphic: Describing two structures (like groups, rings, or graphs) between which an isomorphism exists.

Usage Examples

• Noun:

  • Phép đẳng cấu giữa hai nhóm chứng tỏ chúng có cùng cấu trúc. (The isomorphism between the two groups proves they have the same structure.)
  • Nhà toán học đã tìm thấy một đẳng cấu thú vị. (The mathematician found an interesting isomorphism.)

• Adjective:

  • Hai đồ thị này là đẳng cấu. (These two graphs are isomorphic.)
  • Các không gian vectơ đẳng cấu có cùng số chiều. (Isomorphic vector spaces have the same dimension.)

Advanced Usage

• In abstract algebra, an đẳng cấu nhóm (group isomorphism) is a bijective homomorphism between two groups.

  • Việc xây dựng một đẳng cấu nhóm thường rất phức tạp. (Constructing a group isomorphism is often very complex.)

• In graph theory, two graphs are đẳng cấu đồ thị (graph isomorphic) if there is a one-to-one correspondence between their vertices that preserves adjacency.

  • Bài toán xác định hai đồ thị có đẳng cấu hay không là một thách thức. (The problem of determining whether two graphs are isomorphic is a challenge.)

Variants and Related Words

• Phép đẳng cấu (n): The isomorphism (the mapping itself).
• Tính đẳng cấu (n): The property of being isomorphic; isomorphism.
• Đẳng cấu với nhau (phrase): To be isomorphic to each other.

Synonyms

• Isomorphism: The most direct and precise synonym.
• Structure-preserving bijection: A descriptive synonym emphasizing the mapping's properties.

Related Concepts

• Đồng cấu (Homomorphism): A structure-preserving map that is not necessarily bijective. Every isomorphism is a homomorphism, but not vice versa.
• Tự đẳng cấu (Automorphism): An isomorphism from a mathematical object to itself.

1. (toán học) Isomorphous